[FOM] CH and mathematics
aa at tau.ac.il
Mon Jan 28 01:20:20 EST 2008
On Fri, Jan 25, 2008 at 09:48:40PM -0500, James Hirschorn wrote:
> On Wednesday 23 January 2008 15:03, Arnon Avron wrote:
> > Let me add that in general I cannot understand an argument
> > that something is true because things will look ?bad if it is not.
> > What a kind of a *mathematical* argument is this?? (and
> > if productivity is the issue rather than meaninmgfulness and
> > truth, than why not accept as true V=L, which is a very
> > fruitful axiom?).
> Productivity cannot be the sole criterion for truth because two incompatible
> statements can both be productive, e.g. V=L and MM (Martin's Maximum).
> The axiom V=L seems particularly easy for a Platonist to refute. I should
> think that any reasonable Platonist believes that the concept of infinity, as
> being beyond the finite, has a reality independent of any formalization. Then
> from Cantor they know that some infinities are larger than others. Along
> these lines, of not putting an artificial "ceiling" on infinities, I suspect
> they believe that large cardinal axioms are true provided that they are
> consistent (at least for the currently known LC axioms). However, even the
> existence of a measurable cardinal negates V=L.
> There are surely also other very convincing Platonistic arguments against V=L.
I am not arguing about that. The only point I wanted to make
at the above quoted paragraph is that it is intelectually dishonest
to reject one axiom (V=L) because it seems to be FALSE, but to
argue for another one (-CH) because there are reasons to view it as
more productive than its negation. Even from Platonists coherence
should be expected (as long as we are talking about
MATHEMATICS and not Theology).
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